This article is reprinted with the permission of the
International Review of Law and Economics.

International Review of Law and Economics (1982), 2
(81-93) (c) 1982 Butterworths

WHAT IS 'FAIR COMPENSATION' FOR DEATH OR INJURY?

DAVID FRIEDMAN

University of California,

Los Angeles CA 90024, USA

I. INTRODUCTION

The obvious principle for determining fair compensation in injury
suits is that the injuror must 'make good' the damage to the injuree;
that is to say, that he must make him as well off as if the injury
had not occurred.[1] This
principle seems to fit both our intuitions about justice and the
economic view of the legal process, according to which such suits
provide potential injurers with appropriate incentives not to impose
risks on others, by making the injurers bear the cost of their
acts.[2]

Unfortunately, there are some injuries for which the application
of that principle seems difficult or impossible. There may be no
payment large enough to make a blinded man as well off as if the
injury had not occurred. If the injury is fatal the problem is
inherently insoluble, save for those altruistic victims who would
have been willing to trade their lives for a sufficiently large post
mortem donation to their favorite charity. Even where it is possible,
by enormous payments, to just barely recompense the victim of
personal injury, it is not immediately obvious that the creation of
blind billionaires living in profligate luxury (and so compensating,
with pleasures that large amounts of money will buy, for the lost
pleasures that it cannot buy) is a good idea.

The purpose of this article is to argue that 'full compensation'
in the sense just described would in fact be overcompensation, in
terms both of justice and of economic efficiency, and that there is
another criterion which leads to more appropriate levels of
compensation and is better able to deal with the 'impossible' problem
of fairly compensating the victim of death or serious injury. Part II
is a verbal sketch of the arguments and the criterion they lead to;
in Part III I introduce and employ the formal concepts of Von
Neumann-Morgenstern utility and economic rationality to redo the
argument in a more precise way. Part IV summarizes the results and
discusses the relation of my arguments and conclusions to
conventional legal principles.

I have tried to put my arguments in a form accessible to both
economists and non- economists interested in the law; I apologize in
advance to those economists who find my explanations of elementary
economic concepts superfluous and to those non-economists who find
them insufficient.

II. THE ARGUMENT

Compensation for death or bodily injury involves two quite
different problems. The first is the problem of how much' damage
there is to make up for. The second is the problem of in what coin
damages can be paid. One might imagine that someone would be willing
to give his life in exchange for a sufficiently high price--five
years of ecstasy, perhaps. Faust, after all, traded not merely life
but eternal bliss for a finite payment. More mundanely, we observe
that people are willing to enter dangerous professions (driving
dynamite trucks, for example) in exchange for somewhat higher pay,
thus in effect trading life--a small increase in the probability of
getting killed--for income. Both examples suggest that the reason it
is impossible to 'fully compensate' someone for the loss of his life
is not that the value of his life to him is infinite--it is not--but
that the value of compensation to a corpse is in most cases
small.[3] The same
argument applies to less-than-lethal injuries. If an injury somehow
made the victim blind half the time--on alternate weeks--it might be
possible to compensate him by a payment of a million dollars. It does
not follow that two million, or even three million, would be fair
compensation for making someone blind all of the time. In part this
is because being blind all of the time is more than twice as bad as
being blind half the time. But in part it is because in the first
case, some of the payment would be spent in forms of consumption
(travel, opera) which require sight to be of much value; with the
possibility of such forms of consumption eliminated, larger sums must
be spent on other and less attractive forms of consumption in order
to provide the same value to the victim. Another way of putting the
same point is to say that bodily injury makes the victim worse off in
two ways. It lowers his effective income by reducing his earning
power and imposing costs (a seeing eye dog, hospital bills, etc.). In
addition, it lowers the value to him of any given income by
eliminating ways in which he can spend it. Death is the extreme case;
not only does it lower the victim's income to zero, it simultaneously
reduces to zero the benefit he can get by spending any form of
income--including damage payments.

One thing this argument suggests is that 'full compensation'--a
level of payment for damages which restores the victim to the level
of welfare he had before the injury--is in a sense inefficient. To
see this, imagine that the victim first receives compensation
sufficient to make up for any loss of income from his injury, so that
he can afford to consume exactly the same things as if he had not
been injured. Since he is no longer able to consume some of those
things (color television if he has been blinded), he spends what
previously went for goods he can no longer use on the remaining sorts
of consumption. Since he is spending more than before his accident on
these (say, gourmet dinners) one would expect the value to him of
additional expenditures on these goods to be lower than before.
Before his injury, the last dollar spent on 'color television
services' provided the same benefit as the last dollar spent on
gourmet dinners; had that not been the case, he could (and would)
have improved his welfare by spending more on the one form of
consumption and less on the other. After his injury, he must transfer
the money he previously would have spent buying a television to
buying more (and increasingly less pleasurable) gourmet dinners; it
is for this reason--because he is less able to make use of
income--that he is worse off even if his income is not reduced. But
this also implies that with the same income, the benefit he receives
from the 'marginal' dollar is less. If his compensation is sufficient
to both replace his lost income and provide enough additional income
to compensate for the loss of his vision (supposing that to be
possible), the conclusion holds a fortiori. Since in order to
make up for all the pleasures he can no longer enjoy he must consume
pleasures he can enjoy to a point of near satiation, the value to him
of each additional dollar will be very low indeed. Hence 'full
compensation' involves transferring income from uninjured persons,
who can receive large benefits from each dollar, to injured (and
already partly compensated) persons who receive very small benefits
from each dollar.[4]
Intuitively it seems that although this may be fair it is also
wasteful; it is doubtful that actual courts and juries (at least in
cases where the injurer is a visible human being and not a
corporation or insurance company) make any attempt to push
compensation that far.

The reader may suspect that I am now going to argue that the
proper level of compensation is that which restores the injured party
to his previous level of income without making up for the losses due
to his lessened ability to make use of that income. If so he is
wrong. In terms of the `efficiency' argument I have just sketched
even that level seems too high, since even at that level the value of
a dollar to the injured party is, I have argued, less than to a
similar uninjured person with the same income. On the other hand,
that level of compensation is inadequate as a deterrent to the
injurer, since it understates the cost imposed by his actions.

In order to resolve this puzzle and find a level of compensation
which is both efficient and adequate, we must move from the ex
post situation, in which the injury has already occurred, to the
ex ante situation, in

which the potential victims are subject to some probability of
injury or death, but the damage has not yet occurred. Two things are
worth noting from this perspective. The first is that if the risk is
small, there is probably some sum of money such that the potential
victims would be indifferent between receiving that sum and being
subject to the risk, and having neither the money nor the risk. Hence
although the damage may be enormous or even infinite (in the case of
death) when viewed in terms of ex post compensation, it is
finite and may even be small in terms of ex ante compensation.
The second and closely connected observation is that if people are
going to be compensated for a deadly risk, they would much prefer to
receive their money when the risk does not eventuate and they are
therefore alive to spend it.

One possible conclusion is that those who impose risks should be
required to compensate ex ante those on whom the risks are
imposed. The amount of compensation would be such as to just
compensate for the risk (estimated, perhaps, from the risk premium on
hazardous jobs), and the potential victims could then choose how to
allocate the money among the different possible outcomes. If they
believed that the money was more valuable to them if the accident did
not occur they could consume the ex ante damage payments; if
they believed it was more valuable after the accident, they could use
them to buy insurance. If they believed that some but not all of it
would be needed after an accident, they could divide their
expenditures between consumption and insurance accordingly.

Such a solution is unworkable. It would require the courts to
estimate in advance the risks imposed by an enormous variety of
activities, including some for which the very nature of the risk
could not be known until too late and many for which the estimation
of probabilities and potential damages would be difficult for the
parties concerned and virtually impossible for the courts. What we
need instead is a system under which damages are paid when an
accident occurs (at which point the damage done and the fact of the
accident are known) but collected (at least in part) when it does not
occur (that being when the money is of most use to the recipient).
Such a system is by no means impossible.

Imagine that we have a system in which potential victims of
accidents know that they (or their heirs) will be compensated in case
of injury or death; further suppose for the moment that the formula
is 'full compensation'. The potential victim knows that he is no
worse off for being subject to risk; if injured he will be fully
compensated. It might, however, occur to him that full compensation
would involve payments of large amounts of money, much of which (in
his hypothetical injured state) he could make little use of. If he
were sufficiently ingenious and the society sufficiently well
organized, he might then decide to sell insurance on himself. In
exchange for some payment if he is not injured, he would agree to pay
someone else part of the damage payment he receives if he is injured.
Suppose, for example, that the chance of injury were one in a hundred
and the 'full compensation' in case of injury were ten million
dollars. He could agree, in exchange for ten thousand dollars now, to
give the buyer a million dollars if he himself were injured. He would
be left with nine million dollars in case of injury; while that would
not be enough to fully 'make up for' the injury (if it happened he
would wish it had not) the benefit he would expect to get if he 'won'
his bet (i.e. was not injured) would more than make up for the loss
if he 'lost' it, since a certainty of ten thousand dollars when he
was uninjured was worth more to him than one chance in a hundred of
having a million dollars when he was injured (and already compensated
by a nine million dollar payment) .

This argument implies that if potential victims are able to sell
fair insurance on themselves then 'full compensation' is actually
overcompensation. Prior to such a sale, the potential victim is no
worse off through being exposed to risk, since, by the definition of
full compensation, any damage will be made good. After the sale, the
potential victim is better off than before the sale (he has
transferred income from an outcome where it had low marginal utility
to one where it has high marginal utility) hence also better off than
if he were not exposed to risk. But if the potential injurer is
overcompensating, it follows that he will be overdeterred from
imposing risk; he may fail to undertake some risky activities even
though the net benefit more than makes up for the risk. It further
follows from this argument that the correct level of compensation is
that level such that the potential victim, after selling as much
insurance on himself as he wishes, will be neither better nor worse
off than if no risk had been imposed. This is the same result that
would follow from the system of court imposed ex ante damages
discussed above, provided the courts had the necessary knowledge. The
difference is that the injurer pays off when and only when the injury
actually occurs, thus allowing the risk to be measured directly by
ex post outcomes in the real world instead of being estimated
by the court. The potential victim may then transfer income from the
outcome where he is an actual victim to the outcome where he is not
by selling insurance, instead of transferring it the other way by
buying insurance.

While the information problems under this system are reduced, they
are not eliminated. In order for the potential injurer to decide what
risky actions to undertake, he must first estimate the probability
and seriousness of injuries in order to calculate the damages he may
have to pay. He is, unlike the court, in the best possible position
to make such estimates; he is the one taking the actions and
presumably the one who knows most about their consequences. In
addition his welfare depends on making correct estimates; that of the
court may not. In addition to the estimate made by the potential
injurer, a second estimate must be made by the insurance company
which buys insurance on potential victims. Here again, the company
has a private interest in doing a good job; if it overestimates the
risk it will find itself paying more than the insurance is worth and
losing money; if it underestimates it will be outbid by other
companies with better estimates. Unlike the potential injurer, the
insurance company may have no expertise in the particular area,
although it is, unlike a court, expert in the general subject of
risk. And even if the estimates of the insurance companies are wrong,
the result will be only a redistribution between them and their
customers; the actual payments made by the injuror, and hence his
Incentive to avoid risky activities, will be determined by what
happens, not by the insurance companies' estimates.

There remains the question of how much compensation is implied by
this rule. A precise answer depends, as I will show in the next
section, on the size of the risk, something which I have already
argued that the court which must decide on the compensation cannot
know. But as long as the risk is not very large there is an
approximate answer (as I shall also show) which is independent of the
size of the risk. To calculate it one simply takes the sum for which
the potential victim would be willing to accept a very small
probability of death (or injury) and divides by that probability.
Hence if the victim were willing to accept a one in a thousand chance
of death in exchange for a thousand dollars, the damages for actually
killing him should be a million dollars.

Estimating the sum for which the average person would be willing
to accept some small probability of death is difficult but not
impossible. One approach is to estimate the wage premium on dangerous
professions;[5] The
problem with this is that those who enter such professions are
presumably people with abnormally low objections to lethal risks; the
figure calculated in that way will accordingly under-estimate the
figure for the average victim. Other and more indirect ways might
involve the observation of choices concerning amount and quality of
medical care (where the purchaser has some grounds for estimating the
effect on life expectancy), speed of driving, or other choice
variables which affect probabilities of death or injury.

III. THE MATHEMATICS

A utility function is a formal way in which economists describe
how the attractiveness of alternative outcomes affects decisions. It
may be thought of as a numerical measure of how much an individual
values various alternatives, and expressed as a numerical function of
variables such as health, goods consumed, etc. The function is so
constructed that if the individual prefers one outcome (consuming 50
pounds of steak a year, working 40 hours a week, and living to 90) to
another (40 pounds, 35 hours, 95), then the utility of the first set
of values (for the variables 'consumption of steak', 'hours worked',
'lifetime') is higher than that of the second.

It is often convenient to think of a utility function as separated
into several different parts, each depending on a different variable.
Thus one can think of my total utility as being made up of the
utility I get from reading books plus the utility I get from playing
with my children, plus. . . . Such a description is a simplification
of real utility functions, since preferences with regard to one form
of consumption are likely to depend on how much I am consuming of
something else; nonetheless the simplification is often a useful one.
In considering the particular issue of injury, it may also be useful
to separate effects on income from effects on consumption by
imagining that the income you earn is itself a function of several
inputs, among them the possession of certain innate abilities such as
the ability to see. Your utility is then a function of your abilities
and of other inputs, many of which can be purchased with
income.[6] The individual
seeks to maximize his utility subject to a budget constraint:
expenditure on consumption goods equals income (I neglect, for
purposes of simplicity, the fact that saving and borrowing may be
used to reallocate income across time). The effect of injury may then
be usefully divided into its effect on the victim's ability to
produce income and its effect on his ability to use income to produce
utility (for himself). The latter may be further simplified by
supposing that the injury affects the victim's utility function in
two ways.

1. It requires him to spend a certain amount of money
to buy substitutes for things that he would have had at a lower cost
or for free before (purchasing a seeing eye dog, for example). This
can be considered as equivalent to a further reduction in his ability
to produce income; some of the income he produces must be diverted to
purchase these things, and only what is left is available for
ordinary consumption goods.

2. It eliminates certain consumption possibilities, certain of the
ways of converting income into utility (seeing movies or watching
television--in the case of blinding) that went into the (assumed)
additive utility function. By mak ing these assumptions we get a
reasonably simple but not totally unrealistic model for illustrating
the arguments of the previous section; it may be written as follows :

Here x is a vector of consumption goods; xl is
the quantity of good 1 consumed, x2 is the quantity of
good 2 consumed, and so forth. p is the corresponding vector
of prices; hence xp, defined as xl times
p1 plus x2 times p2 plus . . . , is
the total amount spent on consumption goods (the amount spent on good
1 plus the amount spent on good 2 plus. . .). a is a vector of
abilities (to see, to walk, to speak, etc.); each activity requires a
corresponding ability. Y is income (net of any special expenditures
required by an injury) and depends on abilities.

To analyze the effect of an
injury[7] I suppose that
there are only two abilities, al and a2, that
the injury eliminates the first of them (al= 0) and so
makes it impossible to get utility from good 1, and that the
elimination of that ability also lowers income by an amount b
(including both direct and indirect effects). In addition, I set
p1 and p2 equal to one by defining my unit of
quantity as `one dollar's worth'. More complicated situations (in
which there are more abilities, in which several abilities enter each
form of consumption, in which the individual's time is an input to
both his income and utility functions and must be divided between the
two, and in which the elimination of an ability alters the form of
the income function) could of course be considered.

One more element is needed before this model can be used to repeat
part of the argument of the previous section. We must suppose that as
consumption of any good increases, the additional benefit received
from an additional unit (`marginal utility of the good') declines. In
other words, the greater xl is, the less the additional
utility from an additional unit of xl, and similarly for
x2. Prior to the injury, the individual adjusts the
quantities he consumes so as to maximize his utility by shifting
consumption between xl and x2 until the
additional utility he receives from an additional dollar's worth of
each good is the same; as long as the marginal utilities are
different, he can increase his total utility by spending one more
dollar on the good with the higher marginal utility, and one less on
the good with the lower marginal utility.

Let the victim's original income be y. After the injury his income
(assuming no compensation) is y' = y - b (primed variables are
post-injury values). Since he can no longer produce utility via
U1, he spends all of his income on buying x2.
Hence x2'=y'. If this is less than x2 (in other
words, if the reduction in his income plus the additional expenses
imposed by the injury is more than what he used to spend on forms of
consumption which are no longer available to him) the marginal
utility he gets from a dollar after the injury is more than the
marginal utility from a dollar before his injury. If x2'
is greater than x2 the opposite is true; money is worth
less to the victim after his injury than before since the injury has
reduced his ability to use income more than it has reduced the income
he has to spend.

Now suppose that the court compensates the victim by awarding him
damages of b (since b is an income, one should think of the award as
a sum large enough to yield an income of b dollars per year for the
period of the injury--the rest of his life if it is permanent). His
income, after paying for costs imposed by the injury, is now at its
previous level; y'=y. Since he can only spend the income on one good,
x2' =y' =y=xl +x2. His income is the
same as before but his utility is lower. With his previous pattern of
expenditure, the marginal utility of a dollar (and hence the marginal
utility of one more unit of the good) was the same whichever good it
was spent on. Imagine that he moves to his new pattern by reducing
his consumption of xl by one unit and increasing his
consumption of x2 by one unit, then doing the same thing
again and again until xl is reduced to zero. The marginal
utility of a unit of xl rises (as he consumes less units)
and the marginal utility of a unit of x2 falls (as he
consumes more units). Since for the first unit transferred the
marginal utilities were the same, for each successive unit the
utility gained by additional consumption of x2 is less
than the utility lost by reduced consumption of xl. Hence
his utility when using the same income as before to consume
exclusively x2 is less than it was when that income was
divided between the two goods. To put the argument differently, his
previous pattern of expenditure maximized his utility, given that he
had the option of getting utility from either good. Before the
accident he could have consumed only good 2; he chose not to because
he got more utility by consuming some of each good. After the
accident he is forced to make the choice which he had rejected when
he was free to choose. Hence his utility is lowered.

In the previous case (no compensation) utility was lowered, while
the marginal utility of a dollar might either be raised (if he lost
more income than he had spent on good 1) or lowered. In this case,
since he is consuming more of good 2 than before, both his marginal
utility and his total utility are lowered. To an economist this may
at first seem paradoxical; we normally expect that the same change
(an increase in income) which lowers marginal utility of income also
raises total utility.

Is this level of compensation satisfactory? Not, it would appear,
if our objective is to avoid giving money to people who have little
utility for it. From that standpoint, it is too much compensation.
Nor is it satisfactory if our objective is to adequately deter those
who impose risks, by forcing them to make up for the damage they do.
By that criterion it is too little compensation. We have here the
same puzzle already presented in Part II.

To solve the puzzle we must again switch to the ex ante
view, in which potential victims are seen as facing a lottery of
outcomes which includes some probability of injury. We may then use
an extension of the idea of utility due to Von Neumann and
Morgenstern.

Von Neumann-Morgenstern utility not only allows us to consider
behaviour under uncertainty, it also eliminates a weakness in the
concept of the utility function as I have so far described it. I have
spoken of utility in quantitative terms and described one change as
producing a larger change in utility than another. But operationally,
utility is defined in terms of choices; while we can observe that
someone prefers A to B and B to C, we cannot tell by his choices
whether his preference of A over B is more or less than his
preference of B over C; in the language of utility, we cannot compare
[U(A) - U(B)] to [U(B)- U(C)].

By expanding the idea of utility to cover behaviour under
uncertainty, we can eliminate this ambiguity. Suppose I can choose
either a certainty of outcome B or a lottery with equal chances of A
and C. If I choose the lottery I am, in effect, saying that the
chance of getting A instead of B (which I could have had for certain)
more than makes up for an equal chance of getting C instead of B. In
terms of utility, this translates into the statement that the utility
gain from getting A instead of B is greater than the utility loss
from getting C instead of B, or in other words that U(A) - U(B) >
U(B) - U(C).[8]

More generally, Von Neumann and Morgenstern showed that if an
individual's behaviour under uncertainty meets certain rather weak
consistency
requirements[9] it is
possible to assign a utility to each outcome and describe his
behaviour as choosing whatever lottery maximizes expected
utility.[10] If we
accept the idea that an individual is `equally well off' under either
of two alternatives to which he is indifferent--if in other words we
accept, as economists usually do, an individual's choices as a proper
measure of the benefits he receives from different alternatives-- it
seems natural to say that a lottery with a given expected utility is
equivalent, ex ante, to a certain outcome with the same
utility. In the case of compensation for damages, we can say that if
an individual has imposed on him a probability p of some injury and
receives a compensation c for this risk (whether he is injured or
not) then if the expected utility of a lottery with probability 1-p
of being uninjured and having c and probability p of being injured
and having c is the same as the utility of being uninjured and not
having c (his situation if the risk were not imposed on him) he is no
worse off than before, and so (ex ante) has not been injured.
Ex post, of course, either the injury does occur and he is
worse off than if the risk was not imposed, or it does not occur and
he is better off (since he still gets c).

We can now describe a fair ex ante compensation--that is to
say, one which leaves the potential victim no worse off ex
ante than if the risk had not been imposed. Using our previous
utility function, letting b again be the loss of income resulting
from the injury and c the compensation, we have, for a fair
compensation c:

Here c1 and c2 are the amounts of the ex
ante compensation which the individual chooses to allocate to
goods 1 and 2 if he is not injured. Or, more generally:

Where utility is written as a function of income and ability.

This gives us a definition of fair compensation (although I have
not yet discussed how it might be turned into real world numbers) but
it is not yet an entirely satisfactory one. Looking at Eq. (1), there
is no reason to expect that the marginal utility of income will be
the same whether or not the injury occurs. Which marginal utility is
larger will depend on whether x1+c1 is larger
or smaller than b. But the individual can transfer income between the
two outcomes by buying or selling insurance on himself; the result of
his doing so will show the sense in which unequal marginal utility of
income in different outcomes is inefficient within the context of Von
Neumann-Morgenstern utility.

Suppose that the individual has access to a fair insurance market.
By giving up a dollar with certainty, he can receive 1/p dollars in
case of injury; alternatively, by giving up a dollar in case of
injury, he can receive p dollars with certainty. Looking at Eq. (2),
it is clear that if the additional utility he receives from a dollar
if the injury does occur is greater than the additional utility he
receives from a dollar if it does not occur, then his expected
utility (the left hand side of Eq. (2)) is increased by buying
insurance. If the inequality goes the other way, he benefits by
selling insurance on himself. Hence a situation in which the two
marginal utilities are unequal is inefficient; the individual, by
buying or selling insurance, can benefit himself without imposing any
cost on others. I assume in this argument that the number of such
individuals is great enough so that the insurer can count on the law
of large numbers to transform the lottery he buys into a nearly
certain outcome with a value equal to the lottery's expected value. I
also (and less plausibly) assume that the transaction costs of
arranging such insurance can be ignored. If such an insurance market
exists, a potential victim receiving the compensation specified by
Eqs (1) and (2) will use it to transfer income from the outcome where
it has the lower marginal utility to the outcome where it has the
higher marginal utility; in doing so he raises his total utility.
Hence that level of compensation, although lower than 'full
compensation', is still too high; the imposition of the risk (plus
the compensation) makes the potential victim better off, ex
ante, than if it had not been imposed. Given such a market the
correct rule for compensation becomes:

Here c is the (ex ante) compensation, c3 is the
amount of it consumed if the injury does not occur (the full
compensation minus the cost of any insurance bought or plus the
amount of any insurance sold), and c4 is the amount
consumed if the injury does occur (c3 plus the amount paid
on insurance bought or minus the amount paid on insurance sold). For
a given value of c, the potential victim is assumed to adjust
c3 and c4 by buying or selling that amount of
insurance which makes the marginal utility of income (the extra
utility from having one more dollar to spend) equal for both of the
utility functions on the left hand side of Eq. (3), and thus
maximizes the left hand side, which is the expected utility given the
probability of injury, the amount of compensation, and the amount of
insurance bought or sold. The final step in the analysis consists of
replacing the ex ante payment (which accompanied the
imposition of risk, whether or not any injury occurred) with an ex
post payment (which occurs only if there is an injury). Since the
hypothetical insurance market permits the potential victim to freely
transfer income between the two outcomes (injury and non-injury) at
an "xchange rate' of p/(l-p) dollars if injured for each dollar if
uninjured, a compensation of c/p if injured is equivalent to a
certain payment of c. Hence our ex post rule is to compensate
the injured party with a payment of c/p, c having the same value as
in Eq. (3).

Before going on to discuss how c/p might be estimated it is worth
seeing how this solution resolves the problems raised in Section I.
First, it adequately compensates the victim and deters the imposer of
the risk; ex ante the victim is no worse off if the risk is
imposed than if it is not. If the injury actually occurs the victim
may and probably will be worse off, but if so he will have
transferred some of his damage payment to the outcome in which the
injury did not occur (by selling insurance on himself) and will be
better off if he is not injured by an amount which, given the
probabilities, just compensates him for his loss if he is.

This solution solves the problem of giving large sums of money to
individuals who cannot benefit from them. Assuming that the potential
victim has correctly allocated the damage payment between alternative
outcomes, the marginal utility of income will be the same to him
whether he is or is not injured. It also deals with most, although
not all, of the situations where 'full compensation' is impossible
due to the inability of the injured (or dead) victim to get
sufficient utility to compensate him from any payment however large.
Under this system the damages may be consumed by potential victims
when they do not become actual victims and are hence able to enjoy
the consumption.

There is still a potential problem. Suppose that the utility of an
uninjured person is bounded; however great his income, he cannot
receive a utility from it of more than Umax=U([[infinity]]
, a). Further suppose that the utility to him of being dead is
U(Y',a')=O. Lastly let the utility of being uninjured be
U(Y,a). If (1 - P) < U(Y,a)/ Umax it is
easily seen that no possible compensation will prevent the potential
victim's expected utility from being lowered by the risk. If
empirical observations of the additional income which people require
in order to accept hazardous jobs are to be trusted, this is likely
to occur only if p is
large.[11] To estimate
c/p, the fair compensation for injury, we find some situation in
which we can observe the payment (c') in exchange for which the
individual is willing to accept a known probability (p') of the same
injury. As I will show below, c/pc'/p', provided that
p and p' are both small.

To see why this is the case, we first note that as long as p is
small, c3 in Eq. (3) must also be small; small amounts of
money spent or received for insurance on a very unlikely event
correspond to large amounts of insurance, and are therefore
sufficient to cause large changes in c4 and
correspondingly large adjustments in the marginal utility of income
(given the injury). The equality of marginal utility of income across
outcomes then gives us:

From this it follows that for small values ofp, c4 is
approximately independent ofp. (U1 is the derivative of
the utility function with regard to income.)

I now use Eq. (3) and the first order Taylor expansion of U about
(Y,a) to get:

Solving for c3 yields:

Substituting this into Eq. (4) gives:

From which it follows that:

This is approximately independent of p for p
small.[12] Hence the
value of c'/p' which is deduced from observed behaviour (the wage
premium on risky jobs, for example) may be used as an approximate
value for c/p, the amount which should be awarded to someone who
suffers the same injury.

IV. THE RESULTS, THE LAW, AND THE REAL WORLD

My solution to the problem of fair compensation depends on the
existence of an insurance market which allows individuals to shift
compensation between different outcomes. No such market appears to
exist; we see individuals purchasing insurance on themselves but not,
as the argument suggests that they ought often to wish to do, selling
it. One explanation is that this is a case of market failure, the
transaction costs for such insurance being too great to allow the
market to exist.[13] If
so, that fact increases the damages that should be paid, substituting
Eq. (2) for Eq. (3).

A second alternative is that the damages currently awarded are too
small, with the result that nobody wants to sell insurance on
himself. Certainly the formulae sometimes used by courts to set
damages in terms of an estimate of the lost income from the injury
could be expected to lead to undesirably low figures.

A third alternative is that there may be legal barriers which make
it difficult or impossible for an individual to sell, in advance, the
damages which he will receive if injured or killed. Since I am an
economist and not a lawyer, that is a possibility about which my
readers may know more than I do. Certainly if such barriers do exist,
my analysis suggests that they should be
eliminated.[14]

Even if my hypothetical insurance market existed, it might be
argued that it would be unfair to limit damage payments to that sum
which would be fair on the assumption that the victim had made use of
that market to buy or sell insurance on himself in order to minimize
the ex ante cost of the risk. I would reply that this
corresponds to the familiar legal doctrine according to which the
liability of the injuror is limited to the damage that would have
occurred if the victim had taken reasonable steps to minimize it; if
the injury caused by your negligence is multiplied by my refusal to
accept treatment, you are not responsible for the result. I admit
that my application of that doctrine is a somewhat unconventional
one.

I will end this paper by saying what I believe that I have and
have not done. I have provided some foundation for the idea (which is
not original with
me[15]) that damage
payments for injury or death ought to be based on the payment which
the individual would require to voluntarily accept a small
probability of the same injury, with damage being set equal to that
payment divided by that small probability. I have also shown that
that formula is not precisely correct; even under the special
circumstances I assumed for my analysis it somewhat understates the
proper damages where the probability of the injury which has occurred
is larger than the probability of the injury used to estimate the
amount of damages; in the absence of those special circumstances it
understates the proper damages substantially. I have therefore not
clearly established what compensation ought to be in the real world
of incomplete markets, but by showing what it should be and how it
might be estimated in a wor of more complete markets I have, I
believe, somewhat clarified the conceptual issues involved in
determining fair levels of compensation.